Baptized In the Name of Jesus
In several places in Acts, the Bible says people were to be baptized in the name of Jesus. And so during baptism ceremonies, those words are spoken in order to follow Biblical principles.
Peter replied, "Repent and be baptized, every one of you, in the name of Jesus Christ for the forgiveness of your sins. And you will receive the gift of the Holy Spirit. (Acts 2:38)
But when they believed Philip as he preached the good news of the kingdom of God and the name of Jesus Christ, they were baptized, both men and women. (Acts 8:12)
When they arrived, they prayed for them that they might receive the Holy Spirit, because the Holy Spirit had not yet come upon any of them; they had simply been baptized into the name of the Lord Jesus. (Acts 8:15-16)
"Can anyone keep these people from being baptized with water? They have received the Holy Spirit just as we have." So he ordered that they be baptized in the name of Jesus Christ. Then they asked Peter to stay with them for a few days. (Acts 10:47-48)
Baptized in the Name of All Three
Yet in chapter 28 of Matthew (in the :great commission) the Bible says people were to be baptized in the name of the Father, Son and Holy Spirit. And some speak these words during the ceremony, partly because it sounds more authoritative and partly because it covers both cases (just in case God might consider one to be one to be better than the other).
Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.
(Matthew 28:19-20)
In His Name
The people who heard these words ("be baptized in the name of ..." ) were Jews. They were leaving behind their old system: Follow the Law to be acceptable to God. The Law included temple worship, offerings and sacrifices and standards for living.
It seemed to provide a way to God's forgiveness and acceptance. Actually, it was only a shadow of what was coming. The reality was Christ Jesus.
The Jews could say His name alone: "baptized in the name of the Messiah Jesus" (calling him Messiah) or in conjunction with the rest. It really didn't matter. The important issue was for them to confess (agree) that Jesus was God --equal with the Father and the Spirit. To say the name of Jesus in their last ceremonial washing caused them to renounce their own efforts and accept his!